Downloaded from orbit.dtu.dk on: Sep 27, 2021 A short introduction to frames, Gabor systems, and wavelet systems Christensen, Ole Published in: Azerbaijan Journal of Mathematics Publication date: 2014 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Christensen, O. (2014). A short introduction to frames, Gabor systems, and wavelet systems. Azerbaijan Journal of Mathematics, 4(1), 25-39. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Azerbaijan Journal of Mathematics V. 4, No 1, 2014, January ISSN 2218-6816 A Short Introduction to Frames, Gabor Systems, and Wavelet Systems Ole Christensen Abstract. In this article we present a short survey of frame theory in Hilbert spaces. We discuss Gabor frames and wavelet frames, and a recent transform that allows to move results from one setting into the other and vice versa. Key Words and Phrases: frames, dual pair of frames, wavelet system, Gabor system 2000 Mathematics Subject Classifications: 42C15, 42C40 1. Introduction Frames provide us with a convenient tool to obtain expansions in Hilbert spaces of a similar type as the one that arise via orthonormal bases. However, the frame conditions are significantly weaker, which makes frames much more flexible. For this reason frame theory has attracted much attention in recent years, especially in connection with its concrete manifestations within Gabor analysis and wavelet analysis. In this article we give a short overview of the general theory for frames in Hilbert spaces, as well as its applications in Gabor analysis and wavelet analysis. Finally, we present a method that allows to construct wavelet frames based on certain Gabor frames, and vice versa. Applying this to Gabor frames generated by exponential B-splines produces a class of attractive dual wavelet frame pairs generated by functions whose Fourier transforms are compactly supported splines with geometrically distributed knots. 2. A survey on frame theory General frames were introduced already in the paper [17] by Duffin and Schaeffer in 1952. Apparently it did not find much use at that time, until it got re-introduced by Young in his book [30] from 1982. After that, Daubechies, Grossmann and Morlet took the key step of connecting frames with wavelets and Gabor systems in the paper [15]. http://www.azjm.org 25 c 2010 AZJM All rights reserved. 26 Ole Christensen 2.1. General frame theory Let H be a separable Hilbert space with the inner product h·; ·i linear in the first entry. A countable family of elements ffkgk2I in H is a (i) Bessel sequence if there exists a constant B > 0 such that X 2 2 jhf; fkij ≤ Bjjfjj ; 8f 2 H; k2I (ii) frame for H if there exist constants A; B > 0 such that 2 X 2 2 Ajjfjj ≤ jhf; fkij ≤ Bjjfjj ; 8f 2 H; (2.1) k2I The numbers A; B in (2.1) are called frame bounds. (iii) Riesz basis for H if spanffkgk2I = H and there exist constants A; B > 0 such that 2 X 2 X X 2 A jckj ≤ ckfk ≤ B jckj : (2.2) for all finite sequences fckg. Every orthonormal basis is a Riesz basis, and every Riesz basis is a frame (the bounds A; B in (2.2) are frame bounds). That is, Riesz bases and frames are natural tools to gain more flexibility than possible with an orthonormal basis. For an overview of the general theory for frames and Riesz bases we refer to [1] and [6]; a deeper treatment is given in the books [2], [4]. Here, we just mention that the difference between a Riesz basis and a frame is that the elements in a frame might be dependent. More precisely, a frame ffkgk2I is a Riesz basis if and only if X 2 ckfk = 0; fckg 2 ` (I) ) ck = 0; 8k 2 I: k2I Given a frame ffkgk2I , the associated frame operator is a bounded, self-adjoint, and invertible operator on H, defined by X Sf = hf; fkifk: k2I The series defining the frame operator converges unconditionally for all f 2 H. Via the frame operator we obtain the frame decomposition, representing each f 2 H as an infinite linear combination of the frame elements: −1 X −1 f = SS f = hf; S fkifk: (2.3) k2I A Short Introduction to Frames, Gabor Systems, and Wavelet Systems 27 −1 The family fS fkgk2I is itself a frame, called the canonical dual frame. In case ffkgk2I is a frame but not a Riesz basis, there exist other frames fgkgk2I which satisfy X f = hf; gkifk; 8f 2 H; (2.4) k2I each family fgkgk2I with this property is called a dual frame. The formulas (2.3) and (2.4) are the main reason for considering frames, but they also immediately reveal one of the fundamental problems with frames. In fact, in order for (2.3) to be practically useful, one has to invert the frame operator, which is difficult when H is infinite-dimensional. One way to avoid this difficulty is to consider tight frames, i.e., frames ffkgk2I for which X 2 2 jhf; fkij = Ajjfjj ; 8f 2 H (2.5) k2I for some A > 0. For a tight frame, hSf; fi = Ajjfjj2, which implies that S = AI, and therefore 1 X f = hf; f if ; 8f 2 H: (2.6) A k k k2I 2.2. Operators on L2(R) 2 In order to construct concrete frames in the Hilbert space L (R); we need to consider some important classes of operators. Definition 2.1. (Translation, modulation, dilation) Consider the following classes 2 of linear operators on L (R): (i) For a 2 R; the operator Ta; called translation by a; is defined by (Taf)(x) := f(x − a); x 2 R: (2.7) (ii) For b 2 R; the operator Eb; called modulation by b; is defined by 2πibx (Ebf)(x) := e f(x); x 2 R: (2.8) (iii) For c > 0; the operator Dc; called dilation by c; is defined by 1 x (D f)(x) := p f( ); x 2 : (2.9) c c c R All the above operators are linear, bounded, and unitary. We will also need the Fourier 1 transform, for f 2 L (R) defined by Z 1 fb(γ) := f(x)e−2πiγxdx: −∞ 2 The Fourier transform is extended to a unitary operator on L (R) in the usual way. 28 Ole Christensen 2.3. Gabor systems in L2(R) 2 2πimbx A Gabor system in L (R) has the form fe g(x − na)gm;n2Z for some parame- 2 ters a; b > 0 and a given function g 2 L (R). Using the translation operators and the modulation operators we can denote a Gabor system by fEmbTnaggm;n2Z: 2πimx It is easy to show that the Gabor system fe χ[0;1](x − n)gm;n2Z is an orthonormal 2 basis for L (R): However, the function χ[0;1] is discontinuous and has very slow decay in the Fourier domain. Thus, the function is not suitable for time-frequency analysis. For the sake of time-frequency analysis, we want the Gabor frame fEmbTnaggm;n2Z to be generated by a continuous function g with compact support. This forces us to consider Gabor frames rather than bases: Lemma 1. If g is be a continuous function with compact support, then • fEmbTnaggm;n2Z can not be an ONB. • fEmbTnaggm;n2Z can not be a Riesz basis. • fEmbTnaggm;n2Z can be a frame if 0 < ab < 1; • For each a; b > 0 with ab < 1; there exists function g 2 Cc(R) such that fEmbTnaggm;n2Z is a frame. In order for a frame to be useful, we need a dual frame. The duality conditions for a pair of Gabor systems were obtained by Ron & Shen [25], [26]. We state the formulation due to Janssen [21]: Theorem 2.2. Given b; α > 0; two Bessel sequences fEmbTnαggm;n2Z and fEmbTnαg~gm;n2Z, 2 2 where g; g~ 2 L (R); form dual Gabor frames for L (R) if and only if for all n 2 Z; X g(x + jα)~g(x + jα + n=b) = bδn;0; a:e: x 2 R: j2Z Theorem 2.2 characterizes pairs of dual Gabor frames, but it does not show how to construct convenient pairs of Gabor frames. A class of convenient dual pairs of frames are constructed in [5] and [9]: 2 Theorem 2.3. Let N 2 N. Let g 2 L (R) be a real-valued bounded function for which supp g ⊆ [0;N] and X g(x − n) = 1: (2.10) n2Z 1 2 Let b 2]0; 2N−1 ]. Define g~ 2 L (R) by N−1 X h(x) = ang(x + n); n=−N+1 A Short Introduction to Frames, Gabor Systems, and Wavelet Systems 29 Figure 1: The generators B2 and B3 and some dual generators. where a0 = b; an + a−n = 2b; n = 1; 2; ··· ;N − 1: 2 Then g and h generate dual frames fEmbTnggm;n2Z and fEmbTng~gm;n2Z for L (R).